numerical methods in finite element analysis

numerical methods in finite element analysis

Numerical Methods in Finite Element Analysis

Introduction

Finite Element Analysis (FEA) is a widely used numerical method

for solving engineering problems. It involves dividing a complex

geometry into simple elements and solving equations to

determine the behavior of the structure under different loading

conditions. Numerical methods are essential in FEA as they

provide efficient and accurate solutions to complex problems.

This article will discuss the numerical methods used in FEA.

1. Basics of Finite Element Analysis

FEA involves dividing a complex geometry into simple elements

such as triangles or rectangles. Each element is defined by a set

of nodes and equations are solved at these nodes to determine

the behavior of the structure. The equations are usually based

on the principle of virtual work, which states that the work done

by external forces on a structure is equal to the internal work

done by stresses within the structure.

2. Types of Elements

There are different types of elements used in FEA, including

linear, quadratic, and cubic elements. Linear elements have

straight edges, while quadratic and cubic elements have curved

edges. The choice of element depends on the complexity of the

geometry and accuracy required for analysis.

3. Numerical Integration

Numerical integration is used to evaluate integrals that arise in

FEA equations. The most commonly used integration methods

include Gauss-Legendre quadrature and Newton-Cotes

formulas such as trapezoidal rule and Simpson's rule.

4. Matrix Manipulation

The equations generated from FEA are usually represented in

matrix form, which requires manipulation using matrix algebra

techniques such as inversion, multiplication, and

addition/subtraction.

5. Solution Techniques

There are different solution techniques used in FEA, including

direct solvers, iterative solvers, and preconditioners. Direct

solvers involve solving the entire system of equations at once

using matrix inversion techniques such as LU decomposition or

Cholesky factorization. Iterative solvers involve solving the

system one equation at a time using iterative methods such as

Jacobi or Gauss-Seidel. Preconditioners are used to improve the

convergence rate of iterative solvers.

6. Boundary Conditions

Boundary conditions are essential in FEA as they define the

behavior of the structure at its boundaries. The most commonly

used boundary conditions include fixed boundary conditions,

which prevent movement in a particular direction, and

symmetry boundary conditions, which assume that the structure

is symmetric about a plane.

7. Mesh Generation

Mesh generation is the process of dividing a complex geometry

into simple elements for FEA analysis. There are different mesh

generation techniques, including structured meshing,

unstructured meshing, and adaptive meshing.

8. Error Estimation

Error estimation is essential in FEA as it provides an indication of

the accuracy of the solution. The most commonly used error

estimation techniques include residual-based error estimation

and recovery-based error estimation.

Conclusion

In conclusion, numerical methods are essential in Finite Element

Analysis as they provide efficient and accurate solutions to

complex engineering problems. The choice of numerical

method depends on the complexity of the geometry and

accuracy required for analysis. Understanding these methods is

crucial for successful application of FEA in engineering design

and analysis.